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5. Pore-Network Modeling of Two-Phase Flow
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where µ is fluid viscosity, k is the network intrinsic permeability, A is the network
cross-sectional area, L is the pore-network length in flow direction, and
∆P is the pressure difference between the inflow and outflow reservoirs.
Repetition of this process at consecutively larger imposed capillary pressures
results in a graph of capillary pressure versus saturation and relative permeability
versus saturation. Results are presented in Section 5.4.
5.4 Results
5.4.1 Flow field in the MDPN model
The calculation of relative permeability is based on computing local velocity
fields within individual pore elements of the network. Thus, the distribution of
velocity field within the network has a crucial role in determining the (relative)
permeability of the porous medium. Capturing an accurate velocity field can
help to calculate more accurate permeabilities. In this section, we show the
advantage of MDPN model in capturing a more accurate flow field.
Compared to the MDPN, a regular network with a fixed coordination number
of six has connections only in three principal directions. In this case, 1/3 of the
connections are perfectly parallel to the overall flow direction, while the rest
are completely perpendicular to the overall flow direction. Since 67% of the
connections are perpendicular to the flow direction, the overall flow direction
may not be the principal direction of the conductivity tensor. Also, the pore
throats which are parallel to the flow direction will have much higher velocities
compared to the pore throats perpendicular to the flow direction. This is
because parallel pore throats form a continues path from the inlet boundary
all the way to the outlet boundary. To illustrate this, we made a network with
connections only in direction numbers 1, 2, and 3 (i.e., a regular pore network
with connections only in three principal directions). After simulating the flow,
we averaged the velocities in each of the three directions to get three average
velocities, one for each direction. These are shown by white columns in Figure
(5.9). It is clear that the average velocity in the direction parallel to the flow
(nearly 0.04 m/day) is much larger than the average velocities in the other two
directions (about 0.015 m/day). Figure (5.10a) shows the scatter diagram of
velocities within the pores of the regular network, which also shows that pore
throats parallel to the flow direction have higher velocities. We also generated
a MDPN model which had connections in all 13 directions, and we averaged
velocities in each direction, resulting in 13 average velocities, also shown in
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